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Several threads (e.g. Integration of the product of pdf & cdf of normal distribution Integration of the product of pdf & cdf of normal distribution ) have shown that

$E[\Phi(x)]=\Phi(\mu/\sqrt{\sigma^2+1})$ when $x\sim N(\mu,\sigma^2)$.

I'd like to compute $Var(\Phi(x))$ for such an $x$, ideally without numerically integrating. Does anyone have any ideas?

Several threads (e.g. Integration of the product of pdf & cdf of normal distribution ) have shown that

$E[\Phi(x)]=\Phi(\mu/\sqrt{\sigma^2+1})$ when $x\sim N(\mu,\sigma^2)$.

I'd like to compute $Var(\Phi(x))$ for such an $x$, ideally without numerically integrating. Does anyone have any ideas?

Several threads (e.g. Integration of the product of pdf & cdf of normal distribution ) have shown that

$E[\Phi(x)]=\Phi(\mu/\sqrt{\sigma^2+1})$ when $x\sim N(\mu,\sigma^2)$.

I'd like to compute $Var(\Phi(x))$ for such an $x$, ideally without numerically integrating. Does anyone have any ideas?

Post Closed as "Needs details or clarity" by Chris Godsil, Alexey Ustinov, Wolfgang, Alex Degtyarev, Franz Lemmermeyer
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Variance of the normal CDF

Several threads (e.g. Integration of the product of pdf & cdf of normal distribution ) have shown that

$E[\Phi(x)]=\Phi(\mu/\sqrt{\sigma^2+1})$ when $x\sim N(\mu,\sigma^2)$.

I'd like to compute $Var(\Phi(x))$ for such an $x$, ideally without numerically integrating. Does anyone have any ideas?